Method and system for controlling the eccentricity of a near-circular orbit

ABSTRACT

A method and system for controlling the eccentricity of a near-circular orbit are embodied in a burn controller for an orbiting object (such as a satellite) that is configured to control burns to occur at either an apogee or a perigee of the orbit to effect a desired change in the eccentricity of the orbit. These burns occur at either an apogee or a perigee of the orbit depending upon the satellite&#39;s location in the long-period motion of the argument of perigee.

BACKGROUND OF THE INVENTION

Several constellations currently inhabit the LEO environment (OrbComm,Iridium, GlobalStar) and more are in the deployment stages or areplanned for the future (Teledesic, etc.). Some of these constellationstake advantage of orbital dynamics to maintain the same perigee locationin inertial space with a non-varying value of the eccentricity. Theseare called “frozen” orbits; the optimal eccentricity of a frozen orbitis called the “frozen eccentricity”. Frozen orbits can assist in theoperational aspects of a constellation by providing easier and moreconsistent satellite-to-satellite linkage as well as reducing theresidence times of satellite failures.

The problem, however, lies in the initial deployment of the vehicle orsubsequent errors in the maneuvering. For near-circular orbits, thefrozen eccentricity is difficult to achieve during deployment and theonly currently known way to drive the eccentricity to the frozen valueonce a vehicle has been deployed is to either conduct radially directedburns or transfer Hohmann burns in pairs. But radial burns ofteninterfere with operational constraints and pairs of Hohmann burnsrequire extra fuel. Thus, the frozen quality of the orbit is difficultto achieve, and the effectiveness of using the frozen orbit within theconstellation is therefore diminished. Under current stationkeepingalgorithms, changing the eccentricity in near-circular orbits is eithercostly or can only be accomplished by burning in the radial direction.Operational constraints often prohibit burning in these ways.Accordingly, it would be desirable to be able to move the eccentricityto the frozen value without interfering with mission operations, orwithout requiring extra fuel beyond that which is expended throughtypical drag-compensation burns in the along-track direction when thesatellites occupy a regime low enough that drag is the dominantnon-conservative force.

BRIEF DESCRIPTION OF THE DRAWINGS

Detailed description of embodiments of the invention will be made withreference to the accompanying drawings:

FIG. 1 illustrates an orbiting object, a satellite, in a frozen lowearth orbit (LEO);

FIG. 2A illustrates a non-frozen orbit;

FIG. 2B illustrates a frozen orbit;

FIGS. 3A-3C illustrate eccentricity vectors for non-frozen, frozen andideally frozen orbits, respectively;

FIGS. 4A and 4B illustrate eccentricity vector motion for frozen andnon-frozen orbits, respectively;

FIG. 5 illustrates eccentricity vector change;

FIGS. 6A and 6B illustrate eccentricity vectors before and after burn,respectively;

FIG. 7 is an flow diagram of an exemplary bum controller according tothe present invention; and

FIG. 8 illustrates an exemplary eccentricity vector control burnscenario according to the present invention.

DETAILED DESCRIPTION

The following is a detailed description of the best presently known modeof carrying out the invention. This description is not to be taken in alimiting sense, but is made merely for the purpose of illustrating thegeneral principles of the invention.

A method and system for controlling the eccentricity of a near-circularorbit according to the present invention are embodied in a burncontroller for an orbiting object (such as a satellite) that isconfigured to control burns to occur at either an apogee or a perigee ofthe orbit to effect a desired change in the eccentricity of the orbit.FIG. 1 illustrates an exemplary operating environment 100 for asatellite 102 in a frozen low earth orbit (LEO).

FIGS. 2A and 2B illustrate differences between a non-frozen and a frozenorbit. With respect to the non-frozen orbit (FIG. 2A), earth oblatenesscauses the perigee to secularly precess, i.e., the perigee rotatesthrough a full cycle in the plane of the orbit. With respect to thefrozen orbit (FIG. 2B), a judicious choice of initial LEO conditionsallows the perigee to be frozen, i.e., the perigee does not rotate, butstays in a confined region of space. Frozen orbits help to facilitategood satellite-to-satellite linkage, altimetry (consistent heightreproduction), and reduced collision risk (“thins” the activeconstellation shell).

According to the present invention, fuel burns are controlled such thatthe eccentricity of the orbit of an object (e.g., a satellite) is movedto the frozen value using drag compensation burns in the along-trackdirection (of the velocity vector associated with the orbiting object)thereby not interfering with mission operations or requiring extra fuelbeyond that ordinarily required to effect drag compensation. In oneembodiment, burn targeting is implemented via a controller that executesa stationkeeping algorithm that uses along-track burns to slowly changethe eccentricity. These burns are conducted in concert with the normalalong-track burns necessary to compensate for the effects of drag. Thus,the change in eccentricity is achieved without any additional fuelconsumption.

FIGS. 3A-3C illustrate eccentricity vectors for non-frozen, frozen andideally frozen orbits, respectively. For near-circular orbits, theeccentricity (e) and the argument of perigee (ω) can be ill-defined asorbital elements. In such situations, it is often more useful toconsider the motion of the eccentricity vector instead of the usualorbit elements. The eccentricity vector is defined as: $\begin{matrix}{\begin{pmatrix}\xi \\\eta\end{pmatrix} = \begin{pmatrix}{e\quad \cos \quad \omega} \\{e\quad \sin \quad \omega}\end{pmatrix}} & (1)\end{matrix}$

Frozen orbits are created through the targeting of the eccentricityvector of a near circular orbit (e.g., eccentricity<0.01) to a certainlocation. At this location, the odd zonal components of the geopotentialcombine with the even zonals to create a stable value of theeccentricity. The vector then stays in the vicinity of this stable valuerevolving about it with a periodicity determined from the secularprecession in the perigee due to J₂: $\begin{matrix}{\omega_{\sec} = {\frac{3}{4}{{{nJ}_{2}\left( \frac{a_{e}}{p} \right)}^{2}\left\lbrack {4 - {5\sin^{2}}} \right\rbrack}}} & (2)\end{matrix}$

where ω is the argument of perigee, p is the semi-latus rectum, n is themean motion, i is the inclination, a_(e), is the Earth's radius, and J₂is the value for the Earth's oblateness.

FIGS. 4A and 4B illustrate eccentricity vector motion for frozen andnon-frozen orbits, respectively. These figures show the variation of theeccentricity vector over a perigee period (˜110 days for a near-polarsatellite at 750 km altitude). The center point of the circle formed bythe eccentricity vector motion is referred to as the frozen eccentricity(depicted as e_(f) in the figures). When the radius of the circleexceeds the value of the frozen eccentricity (the origin O of the phaseplane ζ-η plot lies within the circle), then the vector loses itsstability and starts to cycle throughout the orbit. If it is desired tomaintain the perigee in a confined region in space, then the radius ofthe circle must be held less than the frozen value. Thus, in order toachieve a frozen orbit, the tolerance on the eccentricity vector motion(radius of the circle) must be less than the value of the frozeneccentricity. Cook, G. E., Perturbations of Near-Circular Orbits by theEarth's Gravitational Potential, Planet. Space Sci., Vol. 14, pp433-444, 1966, incorporated herein by reference, gives the formula forthe long period behavior of the eccentricity vector as: $\begin{matrix}{\begin{pmatrix}{\Delta \quad \xi} \\{\Delta \quad \eta}\end{pmatrix}_{LP} = {\begin{pmatrix}{\delta \quad e_{T}{\cos \left( {{\omega_{\sec}t} + \alpha} \right)}} \\{\delta \quad e_{T}{\sin \left( {{\omega_{\sec}t} + \alpha} \right)}}\end{pmatrix} + \begin{pmatrix}0 \\e_{f}\end{pmatrix}}} & (3)\end{matrix}$

where δe_(T) is the initial tolerance (radius of the circle) and α isthe phase angle determined by the initial conditions at time t=0,ω_(sec) is the secular precession rate of the argument of perigee whosemotion is due to the even zonals (J₂ typically dominates), and e_(f) isthe frozen eccentricity defined by: $\begin{matrix}{e_{f} = {{\frac{J_{3}}{2J_{2}}\left( \frac{a_{e}}{p} \right)\sin \quad } + {\frac{4}{3}{\sum\limits_{m = 5}^{odd}\quad {\left( {m - 1} \right)\frac{J_{m}}{J_{2}}\left( \frac{a_{e}}{p} \right)^{m - 2}\frac{F_{m,0,{{({m - 1})}/2}}()}{4 - {5\quad \sin^{2}}}}}}}} & (4)\end{matrix}$

where the J_(m) are the values of the zonal geopotential coefficientsfor the mth zonal, and F is Kaula's inclination function. See, Kaula, W.M., Theory of Satellite Geodesy, Blaisdell Publishing Co., Waltham,Mass, 1966, incorporated herein by reference. Because the frozen valueis solely in the η component, the stable point will occur at an argumentof perigee equal to 90° (or −90°, depending upon the inclination). Theeccentricity vector will thus oscillate about an argument of perigee ofeither ±90° with an amplitude that is driven by the value of thetolerance; as the tolerance goes to zero, the argument of perigee willapproach ±90°.

A gradual approach towards changing the tolerance on the eccentricityvector (δe_(T)) is possible that does not require the vehicle to bereoriented nor does it require any extra deltaV over the expectedstationkeeping budget. In simple terms, if the system is constrained toproduce only transverse (to a radius of the orbit) stationkeeping burns,i.e. drag compensation, then the eccentricity will experience itsgreatest change at apogee and perigee. If the burns are conducted onlyat perigee or apogee, then the eccentricity can be controlled and drivento the frozen value irrespective of other operational issues. FIG. 5illustrates eccentricity vector change. As can be seen from the figure,apogee boosts circularize the orbit and perigee boosts make the orbitmore elliptical.

FIGS. 6A and 6B illustrate eccentricity vectors before and after burn,respectively. The change in the tolerance, Δδe_(T), due to an arbitraryimpulsive burn can be found from the Law of Cosines for before the burn:

δe _(T) ² =e ² +e ² _(f)−2e e _(f) cos θ  (5)

and after the burn:

(δe _(T) +Δδe _(T) )²=(e+Δe)² +e _(f) ²−2(e+Δe)e _(f) cos(θ+Δθ)  (6)

where the Δ's are the changes in the elements resulting from the burn.If the convention for the angle θ is taken to be positive in thedirection of the eccentricity vector motion, then the cosine of θ isequal to the sine of the argument of perigee, ω, and the sine of theangle θ is equal to the opposite of the cosine of the argument ofperigee:

sin ω=cos θ cos ω=−sin θ  (7)

Substituting into Equation 6 and solving to first order yields:

δe _(T) Δe _(T) =eΔe−e _(f) sin ωΔe−ee _(f) cos ωΔω  (8)

The equations relating the change in the orbit elements that result froma small impulsive burn in the eccentricity and argument of perigee are(where the subscripts r,t,w refer to the radial, transverse(along-track), and normal spacecraft-centered coordinate frame):$\begin{matrix}{{{\Delta \quad e} = {{\frac{\sqrt{1 - e^{2}}\sin \quad v}{na}\Delta \quad v_{r}} + {{\frac{\sqrt{1 - e^{2}}}{nae}\left\lbrack {\frac{p}{r} - \frac{r}{a}} \right\rbrack}\Delta \quad v_{t}}}}{{\Delta \quad \omega} = {{\frac{{- \sqrt{1 - e^{2}}}\cos \quad v}{nae}\Delta \quad v_{r}} + {{\frac{\sqrt{1 - e^{2}}}{nae}\left\lbrack {1 + \frac{r}{p}} \right\rbrack}\sin \quad v\quad \Delta \quad v_{t}} - {\frac{r\quad \sin \quad u\quad \cot \quad i}{{na}^{2}\sqrt{1 - e^{2}}}\Delta \quad v_{w}}}}} & (9)\end{matrix}$

where v is the true anomaly, α is the semi-major axis, and u is theargument of latitude.

The corresponding changes in the components of the eccentricity vectorare found from Equation 1: $\begin{matrix}{{{\Delta \quad \xi} = {{\frac{\sin \quad u}{na}\Delta \quad v_{r}} + {\frac{2}{na}\cos \quad u\quad \Delta \quad v_{t}} + {\eta \frac{\sin \quad u\quad \cot \quad }{na}\Delta \quad v_{w}}}}{{\Delta \quad \eta} = {{{- \frac{\cos \quad u}{na}}\Delta \quad v_{r}} + {\frac{2}{na}\sin \quad u\quad \Delta \quad v_{t}} - {\xi \frac{\sin \quad u\quad \cot \quad }{na}\Delta \quad v_{w}}}}} & (10)\end{matrix}$

The relation between the changes in the eccentricity and the componentsof the eccentricity vector is:

eΔe=ξΔξ+ηΔη  (11)

Because radial burns of ten require disruption of mission operations,and because normal burns must be large to make up for thenear-circularity of the LEO satellites, according to the presentinvention transverse burns are employed for changing the eccentricityvector. Also, if it is desired to minimize fuel usage, then the usualtransverse stationkeeping burns are used to change the eccentricityvector. Considering only the transverse burns, Equation 8 reduces fornear circular orbits to: $\begin{matrix}{{\Delta \quad \delta \quad e_{T}} = {\frac{2}{na}\frac{1}{\delta \quad e_{T}}\left\{ {{\xi \quad \cos \quad u} + {\left( {\eta - e_{f}} \right)\sin \quad u}} \right\} \quad \Delta \quad v_{t}}} & (12)\end{matrix}$

Equation 12 implies that if the burns are targeted at specific points inthe orbit, then the tolerance can be brought down at the same time asthe normal stationkeeping burns are conducted. Namely, withoutincreasing the deltaV budget at all and in the normal course of missionoperations, the eccentricity vector tolerance can be decreased to aneffectively arbitrary level. Also, it should be noted that if the burnsare constrained to take place at apogee and perigee, then from Equation9 it can be seen that a purely transverse burn will affect only theeccentricity and not the argument of perigee (sin v=0). In essence,transverse burns at apogee and perigee change the magnitude of theeccentricity vector, but not its direction. If burns are not perfectlyconstrained to take place at apogee and perigee, the principles of thepresent invention are still applicable albeit with a decrease inefficiency.

If only the mean motion of the eccentricity vector is considered, thenan additional substitution involving Equation 3 can be made:$\begin{matrix}{{\Delta \quad \delta \quad e_{T}} = {\frac{2}{na}{\cos \left( {u - {\omega_{\sec}t} - \alpha} \right)}\Delta \quad v_{t}}} & (13)\end{matrix}$

Since α is a constant phase offset, it can be set to any value with noloss of generality as long as the initial time is chosen to correspondappropriately. Assuming a convenient value, the initial phase angle canbe set to −90° so that a zero value for ωt+α corresponds to a zero valuefor θ. This occurs when the eccentricity vector is at the bottom of itscycle, point A in FIG. 4A. Using the definition for the argument oflatitude and assuming the burns take place at either apogee or perigee,Equation 13 can be expressed as: $\begin{matrix}{{\Delta \quad \delta \quad e_{T}} = {\frac{2}{na}{\cos \left( {\omega - {{\omega_{\sec}t} \pm {90{^\circ}}}} \right)}\Delta \quad v_{t}}} & (14)\end{matrix}$

where the ± sign indicates whether the burn takes place at perigee (+)or apogee (−). Expressions relating the argument of perigee to the timein the eccentricity vector cycle are found by equating the definition ofthe eccentricity vector (Equation 1) to the mean motion of theeccentricity vector (Equation 8): $\begin{matrix}{{{\sin \quad \omega} = {\frac{e_{f}}{e} + {\frac{\delta \quad e_{T}}{e}{\sin \left( {{\omega_{\sec}t} + \alpha} \right)}}}}{{\cos \quad \omega} = {\frac{\delta \quad e_{T}}{e}{\cos \left( {{\omega_{\sec}t} + \alpha} \right)}}}} & (15)\end{matrix}$

where the eccentricity at any point in the cycle is given by the Law ofCosines:

e ²=(δe _(T))²+2e _(f) δe _(T) sin(ω_(sec) t+α)+e _(f) ²  (16)

Combining these expressions gives the argument of perigee solely interms of the frozen eccentricity, the current tolerance, and the time inthe cycle. Substituting yields (and recalling that the phase angle waschosen to be =90°): $\begin{matrix}{{\Delta \quad \delta \quad e_{T}} = {{\mp \frac{2}{na}}\left\{ \frac{{e_{f}\cos \quad \omega_{\sec}t} - {\delta \quad e_{T}}}{e} \right\} \Delta \quad v_{t}}} & (17)\end{matrix}$

where now the minus sign depicts burns at perigee and the plus sign forburns at apogee. Because it is desired to have negative changes in thetolerance, then whether the burn is to take place at perigee or atapogee is determined by the numerator in Equation 17. Perigee burnsshould be conducted when the expression in the brackets is greater thanzero; apogee burns when it is less than zero. The boundary point betweenthese two regimes is simply given when the numerator is equal to zero:$\begin{matrix}{{\cos \quad \omega_{\sec}t} = \frac{\delta \quad e_{T}}{e_{f}}} & (18)\end{matrix}$

Geometrically, this represents the point of maximum deviation for theargument of perigee. This point can also be found in terms of thecomponents of the mean eccentricity vector. The η-component is thevariable of interest since the ξ-component is symmetrical about theη-axis. Therefore when the mean value of the η-component is below acertain value, then the stationkeeping burns are performed at perigee;when it is above this value, drag compensation are conducted at apogee.

FIG. 7 is a flow diagram of an exemplary burn controller 700 accordingto the present invention. The exemplary controller 700 is configured asshown to provide a normal stationkeeping burn to maintain semi-majoraxis (block 702) and determine whether the burn is to be conducted atperigee or apogee to drive the eccentricity to the frozen value (block704).

From the geometry of FIG. 8, the limiting value of the η-component,η_(b), that is equivalent to Equation 18 can be found as:$\begin{matrix}{\eta_{b} = {e_{f}\left\lbrack {1 - \left( \frac{\delta \quad e_{T}}{e_{f}} \right)^{2}} \right\rbrack}} & (19)\end{matrix}$

FIG. 8 also summarizes the eccentricity vector control burn scenario.The normal drag compensation stationkeeping burns are conducted in theusual mode. The difference is that instead of doing the burns at randomlocations in the orbit as is usually done, the burns are targeted ateither apogee or perigee (depending upon the mean eccentricity vectormotion) to affect a desired change in the eccentricity. When the meaneccentricity vector is at the bottom of its perigee period cycle, thedrag compensation burns are conducted at perigee; when the mean vectoris at the top of its cycle, burns occur at apogee. In this manner, theeccentricity vector tolerance is slowly decreased as the normalstationkeeping burns are happening.

Because the usual stationkeeping burns are being used to effect thedecrease in the tolerance, the change in the tolerance under thisscenario will be very gradual. In order to discern how much time itwould take for this modified stationkeeping strategy to implement achange in the eccentricity vector tolerance, Equation 17 is integratedover a perigee period. The total amount that the tolerance will decreaseover the perigee period for a sample satellite in a 750 km altitudenear-circular, near-polar orbit is (perigee period ˜110 days):

(Δδe _(T))_(total)=1.705×10⁻⁴ <Δv _(t)>  (20)

where <Δv_(t)> is the average value of the normal stationkeeping deltaVover that same perigee period in m/sec. For example, at the time of peaksolar activity when drag is greatest, the sample satellite can expect torequire about 4 m/s of stationkeeping deltaV to counteract the effectsof drag per year. This translates to an average stationkeeping deltaVover the 110.5 day perigee period of 1.21 m/sec, in turn giving adecrease in the eccentricity vector tolerance of 2×10⁻⁴. The minimumstationkeeping deltaV occurs at the minimum of the solar cycle and willbe about 0.15 m/sec for that year; this translates to an average perigeeperiod deltaV of 0.045 m/sec and a decrease in the tolerance of 8×10⁻⁶.This implies that in order to reduce a sample tolerance of 0.00074 downto a near-zero value, then 3-4 perigee periods (about 1 year) would berequired during the time of solar maximum, while the benefit gained inthe tolerance is virtually non-existent during the times of solarminimum.

Thus, according to an embodiment of the present invention, theeccentricity vector is controlled and driven closer to the frozen valuewithout any increase in the usual fuel budget by conducting thedrag-compensation burns at apogee or perigee depending upon the locationof the satellite in its long period motion of the perigee.

Although the present invention has been described in terms of theembodiment(s) above, numerous modifications and/or additions to theabove-described embodiment(s) would be readily apparent to one skilledin the art. It is intended that the scope of the present inventionextends to all such modifications and/or additions.

I claim:
 1. A method for controlling the eccentricity of a near-circularorbit, comprising: targeting stationkeeping burns for a object in a lowearth orbit (LEO) to occur only at either an apogee or a perigee of theorbit depending upon motion of an eccentricity vector that points to aposition in the LEO rather than to the sun.
 2. A method for controllingthe eccentricity of a near-circular orbit, comprising: controlling burnsfor an object in an orbit to occur at either an apogee or a perigee ofthe orbit depending upon mean eccentricity vector motion to effect adesired change in an eccentricity of the orbit.
 3. The method forcontrolling the eccentricity of a near-circular orbit of claim 2,wherein the burns are controlled such that a tolerance on eccentricityvector motion is less than a frozen eccentricity of the orbit.
 4. Themethod for controlling the eccentricity of a near-circular orbit ofclaim 2, wherein the burns are controlled to be along-track with respectto the orbit.
 5. The method for controlling the eccentricity of anear-circular orbit of claim 2, wherein the burns are controlled to betransverse to a radius of the orbit.
 6. The method for controlling theeccentricity of a near-circular orbit of claim 2, wherein the burns arecontrolled such that a magnitude of the eccentricity changes, but notits direction.
 7. The method for controlling the eccentricity of anear-circular orbit of claim 2, wherein the burns are controlled tomaintain a semi-major axis.
 8. The method for controlling theeccentricity of a near-circular orbit of claim 2, wherein the burns arecontrolled to both counteract effects of drag and to drive theeccentricity to the frozen value.
 9. The method for controlling theeccentricity of a near-circular orbit of claim 8, Wherein the burns arecontrolled such that no additional fuel beyond a normal dragcompensation fuel budget is required to drive the eccentricity.
 10. Themethod for controlling the eccentricity of a near-circular orbit ofclaim 2, wherein the object comprises a satellite.
 11. A system forcontrolling the eccentricity of a near-circular orbit, comprising: a bumcontroller for an object in an orbit, the bum controller beingconfigured to control burns to take place at an apogee or a perigee ofthe orbit depending upon components of a mean eccentricity vector todrive an eccentricity of the orbit to a frozen value.
 12. The system forcontrolling the eccentricity of a near-circular orbit of claim 11,wherein the controller is configured to control the burns to bealong-track with respect to the orbit.
 13. The system for controllingthe eccentricity of a near-circular orbit of claim 11, wherein thecontroller is configured to control the burns to be transverse to aradius of the orbit.
 14. The system for controlling the eccentricity ofa near-circular orbit of claim 11, wherein the controller is configuredto control the burns such that a magnitude of the eccentricity changes,but not its direction.
 15. The system for controlling the eccentricityof a near-circular orbit of claim 11, wherein the controller isconfigured to control the burns to maintain a semi-major axis.
 16. Thesystem for controlling the eccentricity of a near-circular orbit ofclaim 11, wherein the controller operates under control of astationkeeping algorithm that controls the burns to both counteracteffects of drag and to drive the eccentricity to the frozen value. 17.The system for controlling the eccentricity of a near-circular orbit ofclaim 16, wherein the stationkeeping algorithm controls the burns suchthat no additional fuel beyond a normal drag compensation fuel budget isrequired to drive the eccentricity.
 18. The system for controlling theeccentricity of a near-circular orbit of claim 11, wherein the objectcomprises a satellite.